Hierarchy of bounds in free orthotropic material optimization: From convex relaxations to Hashin-Shtrikman via sequential global programming

Abstract

We study free orthotropic material optimization for two-dimensional plane-stress compliance minimization with two well-ordered isotropic phases, motivated by the gap between tensors admissible in classical free-material optimization and tensors realizable by composites. To reduce this gap, we construct a hierarchy of realizability-aware admissible sets induced by zeroth-order, Voigt, and Hashin--Shtrikman (HS) energy bounds, moving from convex relaxations to a tighter nonconvex model. In the convex zeroth-order and Voigt settings, the Voigt set is strictly tighter for intermediate volume fractions and coincides with the zeroth-order set at pure-phase endpoints, and the Voigt model reduces to an isotropic variable-thickness-sheet formulation. In the single-loadcase continuum zeroth-order problem, at least one optimal solution can be chosen orthotropic. For HS constraints, we rewrite the bound as a Voigt term minus a nonnegative correction, clarifying strict tightening for interior volume fractions and local nonconvexity. We further prove that the convex hull of the HS feasible set equals the Voigt set and derive reduced formulations via active-constraint analysis and explicit elementwise volume characterization, including reductions specialized to orthotropic effective tensors. In the single-loadcase continuum setting, the HS relaxation is tight with the Allaire--Kohn relaxed problem, attained in the relaxation sense by sequential laminates, whereas in generic multi-loadcase settings it provides a lower bound on optimal compliance over general microstructures. The resulting nonconvex orthotropic HS problem is solved by sequential global programming, and numerical results confirm the predicted compliance hierarchy and show close agreement with finite-rank laminate references.

Figures (17)

• Figure 1: Comparison of the admissible sets $\mathcal{A}^{(0)}(0.5)$ (zeroth-order, yellow) and $\mathcal{A}^{(1)}(0.5)$ (Voigt, blue). The sets are shown in a selected three-dimensional projection of $\bm{\mathbf{E}}_i$ as outer-envelope boundary surfaces, obtained from $750$ sampled strains $\bm{\mathbf{\varepsilon}}$ with Frobenius norm $\sqrt{2}/2$. Material parameters: $\kappa^- = 0.714\times10^{-9}$, $\kappa^+ = 0.714$, $\mu^- = 0.385\times10^{-9}$, and $\mu^+ = 0.385$.
• Figure 2: Comparison of the admissible sets $\mathcal{A}^{(2)}(0.5)$ (Hashin--Shtrikman, green) and $\mathcal{A}^{(1)}(0.5)$ (Voigt, blue). The sets are shown in a selected three-dimensional projection of $\bm{\mathbf{E}}_i$ as outer-envelope boundary surfaces, obtained from $750$ sampled strains $\bm{\mathbf{\varepsilon}}$ with Frobenius norm $\sqrt{2}/2$. Material parameters: $\kappa^- = 0.714\times10^{-9}$, $\kappa^+ = 0.714$, $\mu^- = 0.385\times10^{-9}$, and $\mu^+ = 0.385$.
• Figure 3: Comparison of the product-space feasible sets $\mathcal{F}_{\mathrm{HS}}$ (green) and $\mathcal{F}_{\mathrm V}$ (blue), defined in Lemma \ref{['lem:HS_convex_hull']}. The figure shows projections in $(\bm{\mathbf{E}}_i, v_i^+)$ for all pairs of selected components of $\bm{\mathbf{E}}_i$ together with $v_i^+$. Each plotted set is shown by its boundary surface, obtained by sampling $v_i^+$ and taking the outer hull of feasible points, using $250$ sampled strains $\bm{\mathbf{\varepsilon}}$ with Frobenius norm $\sqrt{2}/2$. Material parameters: $\kappa^- = 0.714\times10^{-9}$, $\kappa^+ = 0.714$, $\mu^- = 0.385\times10^{-9}$, and $\mu^+ = 0.385$.
• Figure 4: Comparison of the admissible set $\mathcal{A}^{(2)}(0.5)$ (Hashin--Shtrikman, green boundary) and orthotropic sequential-laminate tensors that attain the Hashin--Shtrikman energy bound (pink points). The sets are shown in a selected three-dimensional projection of $\bm{\mathit{E}}$. For fixed $v^+=0.5$, the Hashin--Shtrikman boundary was generated using $750$ sampled strains $\bm{\mathbf{\varepsilon}}$ with Frobenius norm $\sqrt{2}/2$. The realizing laminate tensors were generated by sampling $5{,}000$ stresses $\bm{\mathbf{\sigma}}$. Material parameters: $\kappa^- = 0.714\times10^{-9}$, $\kappa^+ = 0.714$, $\mu^- = 0.385\times10^{-9}$, and $\mu^+ = 0.385$.
• Figure 5: Cantilever beam: geometry and boundary conditions

Theorems & Definitions (44)

• Remark 3.1: Orthotropic restriction
• Proposition 3.1: Orthotropic minimizer for the single-loadcase continuum ZO problem
• proof
• Proposition 3.2: Strict feasible-set inclusion of V-FMO in ZO-FMO
• proof
• Proposition 3.3: Existence of isotropic minimizers for V-FMO (discrete and continuum)
• proof
• Proposition 3.4: Nonnegativity and equality cases of the HS correction term
• proof
• Lemma 3.1: Nonconvexity of the HS-feasible set